The paper treats the problem of perturbed infinite dimensional discrete time linear systems given by x p i+1 = Ax p i + B p u i for every i ∈ N and x 0 ∈ L 2 (Ω), where B p is a bounded operator representing the perturbation affecting the system. We suppose that just a party ω of Ω is controlled. We seek to characterize the set of all variations ω p of ω due to negligent disturbances such that the effect is under a threshold chosen previously. Practical algorithm with simulations are given.